Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dvdmsscn | Structured version Visualization version GIF version |
Description: 𝑋 is a subset of ℂ. This statement is very often used when computing derivatives. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
Ref | Expression |
---|---|
dvdmsscn.s | ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) |
dvdmsscn.x | ⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆)) |
Ref | Expression |
---|---|
dvdmsscn | ⊢ (𝜑 → 𝑋 ⊆ ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | restsspw 16936 | . . . 4 ⊢ ((TopOpen‘ℂfld) ↾t 𝑆) ⊆ 𝒫 𝑆 | |
2 | dvdmsscn.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆)) | |
3 | 1, 2 | sseldi 3899 | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝒫 𝑆) |
4 | elpwi 4522 | . . 3 ⊢ (𝑋 ∈ 𝒫 𝑆 → 𝑋 ⊆ 𝑆) | |
5 | 3, 4 | syl 17 | . 2 ⊢ (𝜑 → 𝑋 ⊆ 𝑆) |
6 | dvdmsscn.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) | |
7 | recnprss 24801 | . . 3 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ) | |
8 | 6, 7 | syl 17 | . 2 ⊢ (𝜑 → 𝑆 ⊆ ℂ) |
9 | 5, 8 | sstrd 3911 | 1 ⊢ (𝜑 → 𝑋 ⊆ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 ⊆ wss 3866 𝒫 cpw 4513 {cpr 4543 ‘cfv 6380 (class class class)co 7213 ℂcc 10727 ℝcr 10728 ↾t crest 16925 TopOpenctopn 16926 ℂfldccnfld 20363 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pr 5322 ax-un 7523 ax-resscn 10786 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-ov 7216 df-oprab 7217 df-mpo 7218 df-1st 7761 df-2nd 7762 df-rest 16927 |
This theorem is referenced by: dvxpaek 43156 etransclem17 43467 etransclem18 43468 etransclem20 43470 etransclem21 43471 etransclem22 43472 etransclem29 43479 etransclem31 43481 etransclem34 43484 etransclem43 43493 etransclem46 43496 |
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